Expression of type Forall

from the theory of proveit.physics.quantum.algebra

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import Conditional, Lambda, M, m, n, x
from proveit.linear_algebra import MatrixMult, MatrixSpace
from proveit.logic import CartExp, Equals, Forall, InSet
from proveit.numbers import Complex, NaturalPos
from proveit.physics.quantum import Qmult

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [m, n], instance_expr = Forall(instance_param_or_params = [M], instance_expr = Equals(Qmult(M), Lambda(x, Conditional(MatrixMult(M, x), InSet(x, CartExp(Complex, n))))), domain = MatrixSpace(field = Complex, rows = m, columns = n)), domain = NaturalPos)

expr:

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\forall_{m, n \in \mathbb{N}^+}~\left[\forall_{M \in \mathbb{C}^{m \times n}}~\left(\left[M\right] = \left[x \mapsto \left\{M \thinspace x \textrm{ if } x \in \mathbb{C}^{n}\right..\right]\right)\right]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	34, 45
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 11
6	Operation	operator: 9 operands: 10
7	Literal
8	ExprTuple	11
9	Literal
10	ExprTuple	12, 13
11	Lambda	parameter: 39 body: 14
12	Operation	operator: 37 operands: 15
13	Operation	operator: 37 operands: 16
14	Conditional	value: 17 condition: 18
15	ExprTuple	34, 19
16	ExprTuple	45, 19
17	Operation	operator: 20 operands: 21
18	Operation	operator: 37 operands: 22
19	Literal
20	Literal
21	ExprTuple	23, 24
22	ExprTuple	39, 25
23	Operation	operator: 26 operand: 39
24	Lambda	parameter: 40 body: 29
25	Operation	operator: 30 operands: 31
26	Literal
27	ExprTuple	39
28	ExprTuple	40
29	Conditional	value: 32 condition: 33
30	Literal
31	NamedExprs	field: 44 rows: 34 columns: 45
32	Operation	operator: 35 operands: 36
33	Operation	operator: 37 operands: 38
34	Variable
35	Literal
36	ExprTuple	39, 40
37	Literal
38	ExprTuple	40, 41
39	Variable
40	Variable
41	Operation	operator: 42 operands: 43
42	Literal
43	ExprTuple	44, 45
44	Literal
45	Variable